Multivariate Calculus 10B, Problem 1

Calculate the following integrals

a) $\int _{0}^{1}\int _{y}^{1}e^{\frac {y}{x}}~dxdy$

b) $\int _{0}^{1}\int _{0}^{cos^{-1}(y)}e^{2x-y}~dxdy$

solution:

a

Step 1:
Here we use change of variable, $\int _{0}^{1}\int _{0}^{x}e^{\frac {y}{x}}~dydx=\int _{0}^{1}[{\frac {1}{x}}e^{\frac {y}{x}}/right|_{y=0}^{y=x}$